Comments on employing the Riesz-Feller derivative in the Schrödinger equation
1 Department of Mathematics, Faculty of Science, Kuwait University, 13060 Safat, Kuwait
2 Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany
Received: 3 June 2013
Revised: 6 August 2013
Published online: 7 October 2013
In this paper, we deal with a fractional Schrödinger equation that contains the quantum Riesz-Feller derivative instead of the Laplace operator in the case of a particle moving in a potential field. In particular, this equation is solved for a free particle in terms of the Fox H-function. On the other hand, we show that from physical viewpoint, the fractional Schrödinger equation with the quantum Riesz-Feller derivative of order α, 0 < α ≤ 2 and skewness θ makes sense only if it reduces to the Laplace operator (α = 2) or to the quantum Riesz fractional derivative (θ = 0). The reason is that the quantum Riesz-Feller derivative is a Hermitian operator and possesses real eigenvalues only when α = 2 or θ = 0. We then focus on the time-independent one-dimensional fractional Schrödinger equation with the quantum Riesz derivative in the case of a particle moving in an infinite potential well. In particular, we show that the explicit formulas for the eigenvalues and eigenfunctions of the time-independent fractional Schrödinger equation that some authors recently claimed to receive cannot be valid. The problem to find right formulas is still open.
© EDP Sciences, Springer-Verlag, 2013